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<div class="add">

<span class="kw">thus </span><a NAME="E1:12_1_1"/>
 <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">v</font>, <font color="Maroon" title="c2">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c1">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">vr</font> = <font color="Maroon" title="c1">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">wr</font> = <font color="Maroon" title="c2">w</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">w</font> = 
<font color="Maroon" title="c3">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">wr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c1">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:12_1_1"/>
 <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">u</font>, <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">ur</font> = <font color="Maroon" title="c1">u</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">vr</font> = <font color="Maroon" title="c2">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">wr</font> = <font color="Maroon" title="c3">w</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E4:12_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">wr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>;<br/>
<a NAME="E5:12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ur</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">wr</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>;<br/>
<a NAME="E6:12_1_1_2"/>
<font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ur</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">vr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>;<br/>
<a NAME="E7:12_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">ur</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">vr</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">wr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:12_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_1_2"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:12_1_1"/>
 <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">vr</font> = <font color="Maroon" title="c1">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a></span>)</span> = 
<font color="Maroon" title="c2">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">v</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:12_1_1"/>
 <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">vr</font> = <font color="Maroon" title="c1">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">w</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">vr</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">w</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> = 
<font color="Maroon" title="c2">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">vr</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:12_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D5" onmouseover="rs('rlvect_1/D5')" onmouseout="rh()">RLVECT_1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">a</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">vr</font> = <font color="Maroon" title="c3">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">wr</font> = <font color="Maroon" title="c4">w</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E4:12_1_1_5"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">vr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>;<br/>
<a NAME="E5:12_1_1_5"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">wr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>;<br/>
<a NAME="E6:12_1_1_5"/>
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">wr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">vr</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">wr</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">vr</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">wr</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1_1_5"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E5:12_1_1_5"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:12_1_1_5"/>
<font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:12_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D6" onmouseover="rs('rlvect_1/D6')" onmouseout="rh()">RLVECT_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">vr</font> = <font color="Maroon" title="c5">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:12_1_1_6"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>;<br/>
<a NAME="E4:12_1_1_6"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_6"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="binop_2.html#D9" onmouseover="rs('binop_2/D9')" onmouseout="rh()">BINOP_2:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:12_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:12_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D7" onmouseover="rs('rlvect_1/D7')" onmouseout="rh()">RLVECT_1:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>  is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">vr</font> = <font color="Maroon" title="c5">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:12_1_1_7"/>
<font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">vr</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:12_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c1">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="convfun1.html#K4" title="CONVFUN1:func.4">RLS_Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">vr</font> = <font color="Maroon" title="c1">v</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">thus </span>1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">v</font> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">vr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D11" onmouseover="rs('binop_2/D11')" onmouseout="rh()">BINOP_2:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c1">v</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
